I'm having a hard time understanding why it's called "chirp". The basis signals aren't chirps (a signal that if you reproduce as sound, sounds like a bird chirp. Frequency increases in time.)
And is there a finite orthogonal basis for the z-transform (of a finite length signal), even?
If I remember correctly, the motivation for the z-transform is that more functions have the transform defined for the z-transform than for the DT Fourier transform. And you less often need generalized functions (Dirac deltas, and their derivatives). But there is no place in which z-transforms show up in signal processing computations, I thought. Since then signals are finite-time and there isn't a need for the convergence benefits of the z-transform.